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I've read about differential forms in the Princeton Companion to Mathematics by Gower and in baby Rudin and I'm having trouble reconciling the two expositions.

Rudin says a k-form in $E$ ($\subset R^n$) is a function $\omega$, symbolically represented by the sum $$\omega = \sum{a_{i_1 \cdots i_k}(x) dx_{i_1}\wedge \cdots \wedge dx_{i_k}}$$

(where the indices $i_1, \ldots, i_k$ range independently from 1 to n), which assigns to each k-surface $\phi$ in $E$ a number $\omega(\phi) = \int_{\phi}{\omega}$ according to the rule $$\int_{\phi}{\omega}= \int_{D}{\sum{a_{i_1 \dots i_k}(\phi(u)) \frac{\partial (x_{i_1}, \ldots, x_{i_k})}{\partial (u_1, \ldots, u_k)} du}},$$

where $D$ is the (compact) domain of $\phi$ (for instance the k-cell $[0,1]^k$) and $\frac{\partial (x_{i_1}, \ldots, x_{i_k})}{\partial (u_1, \ldots, u_k)}$ is the Jacobian.

Meanwhile the article by Tao in the Princeton Companion on differential forms gave me the following understanding: a k-form continuously assigns to each point in $R^n$ a (linear) functional which takes in an infinitesimal k-dimensional parallelepiped with dimensions $\Delta x_1 \wedge \cdots \wedge \Delta x_k $ and returns a scalar. Thus if we evaluate a k-form $\omega$ at some point $x$ we get the linear functional $$\omega_x : \wedge^k (R^n) \rightarrow R.$$ So if $\omega$ is a 0-form then $\omega_x$ sends scalars to scalars, if its a 1-form then it sends (infinitesimal) vectors to scalars, if its a 2-form it sends (infinitesimal) parallelograms to scalars, etc. Finally to integrate a k-form against a k-surface we split up the k-surface into infinitesimal k-parallelepipeds, evaluate the k-form at each base point, plug in the associated k-parallelepiped into the resulting functional, and sum across the entire k-surface. And this process is precisely what Rudin's integral $\int_{\phi}{\omega}$ does.

Please let me know if my understanding is correct. I would greatly appreciate any suggestions about how to better understand differential forms.

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    $\begingroup$ Rudin defines forms in a rather odd way to avoid huge prerequisites (differential manifolds, tangent bundles, tensors). It's a tour de force but a lot of math students will eventually acquire the prerequisites and see forms in a somewhat different light, closer to your description of Tao's treatment (which I'm unfamiliar with). Caveat: the "infinitesimals" you speak of are unknown to me. For a more standard treatment of differential forms there is Lee's Smooth Manifolds, for example. $\endgroup$ – ForgotALot Dec 23 '15 at 6:51

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